System, device, and method for over-current relays protecting motors

ABSTRACT

A system, method, and device for protecting an induction motor are disclosed. The exemplary system may have a module for determining the current drawn by the motor and a module for determining the state of the motor. The system may calculate a used thermal capacity based on a first formula when the motor is in an active state. The system also calculates the used thermal capacity based on a second formula when the motor is in an inactive state. When the used thermal capacity attains a threshold, the relay is tripped thus removing current to the motor and preventing motor from overheating. A method to derive thermal time constants from desired trip time limits (such as those defined by IEC standards or the thermal limit curves provided by motor manufacturers) is also presented. For example, the thermal time constants can then be used in tacking motor used thermal capacity throughout various motor states.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to U.S. Provisional Patent Application No.60/727,464 filed Oct. 17, 2005 entitled “Thermal model based DSPalgorithm for solid-state over current relays protecting AC motors”,which is incorporated fully herein by reference.

TECHNICAL FIELD

The present invention relates to controls and protection for motors andmore particularly, to a device, method, and system for determining thethermal properties of a motor.

BACKGROUND INFORMATION

In current practice, there are two types of control algorithms forsolid-state over-current relays, namely, electromechanical relayalgorithm [1][2] and 1²t algorithm. The electromechanical relayalgorithm is derived from the model of electromechanical relay using thefollowing: $\begin{matrix}{{{{{For}\quad 0} < M < 1}{t(I)}}\quad = \frac{t_{r}}{M^{2} - 1}} & {A1} \\{{{{For}\quad M} > 1}{{t(I)} = {\frac{A}{M^{P} - 1} + B}}} & {A2}\end{matrix}$

Where

-   -   t(I) is the reset time in Eq. (A1) or trip time in Eq. (A2) in        seconds,    -   M is the I_(input)/I_(pickup) (I_(pickup) is the relay current        set point),    -   t_(r) is the rest time (for M=0) defined in IEEE Std        C37.112-1996 Table 1,    -   A, B, constants defined in IEEE Std C37.112-1996    -   p Table 1 to provide selected curve characteristics.

The relay will trip if $\begin{matrix}{{\int_{o}^{T_{P}}{\frac{1}{t(I)}\quad{\mathbb{d}t}}} > 1} & {A3}\end{matrix}$

The discrete form of equation A3 is $\begin{matrix}{{\sum\limits_{k = 0}^{k = n}\frac{\Delta\quad t}{t\left( I_{k} \right)}} > 1} & {A4}\end{matrix}$

Where

-   -   T_(p) trip time in seconds,    -   Δt Sample period in seconds,    -   t(I_(k)) t(I) calculated from Eq. (A1) or Eq. (A2) for k^(th)        sample of current I.

For 0<M<1, t(I_(k)) is a negative number. If the summation in equationA4 keeps going, the sum value will go to—∞. Because this algorithmsimulates the reset dynamics of electromechanical relays, the summationshall be stopped if $\begin{matrix}{{\sum\limits_{k = 0}^{k = n}\frac{\Delta\quad t}{t\left( I_{k} \right)}} < 0} & {A5}\end{matrix}$

This is equivalent to saying that a motor will reach the sameequilibrium temperature without regard to what state the motor isoperating. The algorithm does not take into account whether the motor isoperating at 50% I_(FLA) or at 80% I_(FLA). In reality, a motor reachesdifferent equilibrium temperatures when different currents are suppliedto the motor. Therefore equation A1 does not simulate motor thermaldynamics for 0<M<1, which results in the algorithm not accuratelytracking the used thermal capacity of AC motor under varying load.

The 1²t algorithm uses a locked rotor current I_(LR) and safe stall timet_(LR) as motor thermal limit. The cold trip time t_(trip-C) and the hottrip time t_(trip-H) for currents above pickup are defined by$\begin{matrix}{t_{{trip} - C} = {\left( \frac{I_{LR}}{I} \right)^{2}t_{LRC}}} & {B1} \\{t_{{trip} - H} = {\left( \frac{I_{LR}}{I} \right)^{2}t_{LRH}}} & {B2}\end{matrix}$

The used thermal capacity θ_(n) is calculated by $\begin{matrix}{\theta_{n} = {\frac{\Delta\quad t}{t_{trip}} + \theta_{n - 1}}} & {B3}\end{matrix}$

There is a θ_(hot) or t_(hot) to determine the switch of t_(trip) fromt_(trip-c) to t_(trip-H). Note that equation B3 is usually implementedin DSP and is updated only when M>1. There is no update of θ_(n) for M<1so this algorithm does not consider the cooling effect of 0<M<1.

The Amd2 [3] of IEC 60947-4-2 [4] imposes new thermal memory testrequirements on solid-state relay protecting AC induction motors, whichstate: electronic overload relays shall fulfill the followingrequirements (note table and figures references are provided inAmendment 2 to standard IEC 60947-4-2):

-   -   apply a current equal to Ie until the device has reached the        thermal equilibrium;    -   interrupt the current for a duration of 2×T_(p) (see Table 2 of        [3]) with a relative tolerance of ±10% (where T_(p) is the time        measured at the D current according to Table 3 of [3]);    -   apply a current equal to 7.2×Ie; and    -   the relay shall trip within 50% of the time T_(p).

From the analysis above, it is seen that both electromechanical relayalgorithm and 1²t algorithm may fail the thermal memory test and may notprovide sufficient protection to AC induction motors. Accordingly, aneed exists for a device, method, and system relay algorithm based onmotor thermal model. The algorithm may accurately track a motor's usedthermal capacity when the motor's current varies at any value satisfyingM≧0.

SUMMARY

The present invention is a novel device, system, and method for anover-current relay protecting an induction motor comprising thefollowing actions. The exemplary method may determine the current drawnby the motor. The method may also determine the state of the motor. Themethod may then calculate a used thermal capacity based on a firstformula when the motor is in an active state. The method may alsocalculate the used thermal capacity based on a second formula when themotor is in an inactive state. Once the used thermal capacity attains athreshold the method may trip the relay, remove current from the motor,and prevent the motor from overheating.

The invention may include the following embodiments. In one exemplaryembodiment, the first formula for used thermal capacity is$\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h}}} + {\theta_{n - 1} \times {\left( {1 - \frac{\Delta\quad t}{T_{h}}} \right).}}}$In another exemplary embodiment, the second formula for used thermalcapacity is$\theta_{n} = {\theta_{n - 1} \times {\left( {1 - \frac{\Delta\quad t}{T_{c}}} \right).}}$In other exemplary embodiments, the method may calculate the usedthermal capacity based on a third formula when the motor is in anacceleration state. In yet another exemplary embodiment, the method maycalculate the used thermal capacity based on a fourth formula when themotor is in a deceleration state.

It is important to note that the present invention is not intended to belimited to a system or method which must satisfy one or more of anystated objects or features of the invention. It is also important tonote that the present invention is not limited to the exemplaryembodiments described herein. Modifications and substitutions by one ofordinary skill in the art are considered to be within the scope of thepresent invention, which is not to be limited except by the followingclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages of the present invention will bebetter understood by reading the following detailed description, takentogether with the drawings herein:

FIG. 1 is the operation status of a motor.

FIG. 2 is the difference in Eq. 19 and Eq. 22 versus time.

FIG. 3 are the results of a simulation for IEC thermal memory test.

FIG. 4 is the used capacity of different M values.

FIG. 5 is typical thermal limit curve provided by motor manufacturersfor motors above 250 HP.

FIG. 6 is a flow chart of a standard trip algorithm 600 according to afirst exemplary embodiment of the present invention.

FIG. 7 is a flow chart of a custom trip algorithm 700 according to asecond exemplary embodiment of the present invention.

FIG. 8 is sample trip curves derived from IEC standard 60947-4-2 and itsamendment 2 using algorithm 600.

DETAILED DESCRIPTION

The present invention provides a control algorithm for solid-state relayprotecting for induction motors. The algorithm utilizes a thermal modelof an AC induction motors. The algorithm calculates used thermalcapacity of the motors based on motor currents. The relay trips once theused thermal capacity reaches limit. This invention also provides amethod to derive thermal time constants from motor thermal limit curvesand IEC standard 60947-4-2 [3][4]. The thermal time constants may thenused in the control algorithm 600 and 700 for calculation of the usedthermal capacity. The resultant algorithm may then satisfy therequirements of motor thermal limit curves and IEC standard 60947-4-2[3][4].

The following is a list of nomenclature used throughout the application.

-   -   q heat input to a motor per second, its unit is Joules/sec,        i.e., Watts,    -   dt an infinitely small time interval in seconds,    -   τ temperature rise in ° C., defined as the temperature        difference between the motor and its surroundings,    -   τ_(fin) temperature rise at thermal equilibrium in ° C.,    -   R_(th) thermal resistance in ° C./Watt, defined as the        temperature difference that will cause 1 Watt to flow between        the motor and its surroundings,    -   C_(th) thermal capacitance in Joules/° C., i.e, in        Watts*seconds/° C., defined as the energy required to change the        motor's temperature by 1° C. if no heat is exchanged with its        surroundings (adiabatic process),    -   T_(th) thermal time constant in seconds. If power input and        ambient temperature remain constant, temperature will change 63%        of the remaining excursion in 1 T_(th). T_(th)=R_(th)C_(th),    -   T_(h) thermal time constant used when motor is running with        current inputs, i.e., I>0. In some applications, acceleration        thermal time constant T_(h-acc) is different from running        thermal time constants T_(h-run) for motors above 250 HP,    -   T_(c) thermal time constant used when currents are removed from        the motor, i.e., I=0,    -   I_(FLA) rated full load amperage of the motor,    -   SF service factor of the motor,    -   I_(pickup) pick up current of a relay, I_(pickup)=SF×I_(FLA),    -   M multiples of I_(pickup), M=I/I_(pickup),    -   τ_(max) maximum permissible temperature rise above ambient        temperature of the motor    -   θ used thermal capacity=τ/τ_(max),    -   Δt sampling period of current samples in seconds,    -   I_(LR) locked rotor current,    -   t_(LRC) cold locked rotor safe stall time,    -   t_(LRH) hot locked rotor safe stall time,    -   t_(trip) relay trip time.

FIG. 1 shows the different operational states or conditions of motors.Motor thermal characteristics during states A, B, C, D, and E may besignificantly different. For example, when a totally enclosed fan cooled(TEFC) motor is stopped, the fan is stopped, so heat can not bedissipated as fast as when the motor is running. For motors above 250horsepower (hp), motor thermal characteristics during state B may besignificantly different from those during C and D. An accurate tripcurve derived from the thermal limit curves [5] provided by motormanufacturers may need to be used. First a thermal model of the AC motorwill be described, from which the used thermal capacity and trip timecan be calculated based on thermal time constants. Second,implementation of International Electrotechnical Commission (IEC)standard trip classes using the thermal model algorithm are provided.Third, customized trip curves for motors above 250 HP are discussed.Finally, exemplary flowcharts are provided for applications usingstandard or custom trip curves. Embodiments and aspects of the inventionmay be implemented by a variety of Digital Signal Processing (DSP)devices.

As for any other objects, the first order thermal dynamics of a motor isdescribed by: $\begin{matrix}{{{qdt} - {\frac{\tau}{R_{th}}{dt}}} = {C_{th}d\quad{\tau.}}} & 1\end{matrix}$

For an infinitely small time interval dt, qdt is the heat energy inputto the motor, $\frac{\tau}{R_{th}}{dt}$is the heat energy dissipated from the motor to the surrounding, andC_(th)dτ is the resultant energy change of the motor. Multiplying bothsides of Eq. (1) by R_(th), yields:R _(th) qdt−τdt=R _(th) C _(th) dτ.   2

Substituting T_(th)=R_(th)C_(th) into Eq. (2) gives:R _(th) qdt−τdt=T _(th) dτ.   3

At thermal equilibrium, the heat input and heat loss cancels each other,temperature rise attains a final value τ_(fin), and the motortemperature will cease to rise, i.e., dτ=0° C. Hence, Eq. (3) becomes:R _(th) qdt−τ _(fin) dt=0.   4

Whence:τ_(fin)=R_(th)q.   5

Substituting Eq. (5) into Eq. (3) gives:τ_(fin) dt−τdt=T _(th) dτ.   6

Rearranging Eq. (6) yields: $\begin{matrix}{\frac{dt}{T_{th}} = {\frac{d\quad\tau}{\tau_{fin} - \tau}.}} & 7\end{matrix}$

Integrating both sides of Eq. (7) gives: $\begin{matrix}{\frac{t}{T_{th}} = {{{- \ln}\quad\left( {\tau_{fin} - \tau} \right)} + {k.}}} & 8\end{matrix}$

If at the initial moment t=0, the motor has an initial temperature riseτ₀ above the ambient temperature, then Eq. (8):k=ln(τ_(fin)−τ₀).   9

Substituting for k in Eq. (8), and solving for τ, gives: $\begin{matrix}{\tau = {{\tau_{fin}\left( {1 - {\mathbb{e}}^{- \frac{t}{T_{th}}}} \right)} + {\tau_{0}{{\mathbb{e}}^{- \frac{t}{T_{th}}}.}}}} & 10\end{matrix}$

Since motor thermal resistance may be different at different operatingconditions, different thermal constants may be used in Eq. (10). Whenthe motor is running, the motor can properly dissipate heat andT_(th)=T_(h). When currents are removed, the motor will slow down andeventually stop. When the motor is stopped, the motor heat dissipationcapability may be reduced and T_(th)=T_(c). T_(c) is usually three timesof T_(h) for AC induction motors [7]. The different operating conditionsof motors are shown in FIG. 1. Table 1 shows the variants of Eq. (10)used for different motor status. TABLE 1 Motor Temperature Rise FormulaMotor Status Temperature Rise Formula Description B, C, D$\tau = {{\tau_{fin}\quad\left( {1 - e^{\frac{t}{T_{h}}}} \right)} + {\tau_{o}e^{\frac{t}{T_{h}}}\quad 11}}$obtained by setting T_(th) = T_(h) in Eq. (10). E$\tau = {\tau_{o}e^{\frac{t}{T_{h}}}\quad 12}$ obtained by settingτ_(fin) = 0 and T_(th) = T_(c) in Eq. (10).

The function of a digital overload relay is to translate the currentdrawn by an AC Induction motor into temperature rise τ, and detectwhether this rise of temperature has reached the maximum permissibletemperature rise or not. Therefore, the temperature rise may becorrelated with the current. The temperature rise at thermal equilibriumof the motor is proportional to current square, i.e,τ_(fin)=k₂I²,   13

where k₂ is a constant. If I=SF×I_(FLA)=I_(pickup), thenτ_(fin)=τ_(max)=maximum permissible temperature rise, as indicated in:τ_(max)=k₂I_(pickup) ².   14

Substituting Eq. (13) into Eq. (11) and dividing Eq. (11) and Eq. (12)by Eq. (14) yield: $\begin{matrix}{{\frac{\tau}{\tau_{\max}} = {{\frac{k_{2}I^{2}}{k_{2}I_{pickup}^{2}}\left( {1 - {\mathbb{e}}^{\frac{t}{T_{h}}}} \right)} + {\frac{\tau_{0}}{\tau_{\max}}{\mathbb{e}}^{\frac{t}{T_{h}}}}}},} & 15 \\{\frac{\tau}{\tau_{\max}} = {\frac{\tau_{o}}{\tau_{\max}}{{\mathbb{e}}^{- \frac{t}{T_{c}}}.}}} & 16\end{matrix}$

Since $\frac{\tau}{\tau_{\max}}$is the used thermal capacity θ of the motor at time t, substituting$\frac{\tau}{\tau_{\max}} = {{\theta\quad{and}\quad M} = \frac{I}{I_{pickupx}}}$into Eq. (15) and Eq. (16) yields: $\begin{matrix}{\theta = {{M^{2}\left( {1 - {\mathbb{e}}^{- \frac{t}{T_{h}}}} \right)} + {\theta_{o}{\mathbb{e}}^{- \frac{t}{T_{h}}}}}} & 17 \\{\theta = {\theta_{o}{{\mathbb{e}}^{- \frac{t}{T_{h}}}.}}} & 18\end{matrix}$Given Δt as the sampling period of current samples, the discrete formsof Eq. (17) and Eq. (18) are: $\begin{matrix}{{\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}}}},} & 19 \\{\theta_{n} = {\theta_{n - 1}{{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}.}}} & 20\end{matrix}$

To implement Eq, (19) and Eq. (20) in DSP,${\mathbb{e}}^{- \frac{\Delta\quad t}{T_{th}}}$can be pre-calculated and stored as a constant. A simplified form ofthese two equations can also be used by noting that when$\frac{\Delta\quad t}{T_{th}}$is small enough, and $\begin{matrix}{{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}} \approx {1 - {\frac{\Delta\quad t}{T_{th}}.}}} & 21\end{matrix}$

Eq. (21) is derived from a Taylor series. Substituting Eq. (21) into Eq.(19) and Eq. (20) gives: $\begin{matrix}{\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h}}} + {\theta_{n - 1} \times {\left( {1 - \frac{\Delta\quad t}{T_{h}}} \right).}}}} & 22 \\{\theta_{n} = {\theta_{n - 1} \times {\left( {1 - \frac{\Delta\quad t}{T_{c}}} \right).}}} & 23\end{matrix}$

where θ_(n) is the used thermal capacity at n^(th) sample, θ_(n-1) isthe used thermal capacity at (n-1)^(th) sample, and M_(n) is the M atn^(th) sample. Because motors may be running for days or months withoutstopping, it is very important that the error introduced in Eq. (22) andEq. (23) by the approximation will not diverge as ΣΔt→∞. FIG. 2 showsthe difference between Eq. (19) and Eq. (22) converges to 0 as ΣΔt→∞.Therefore Eq. (22) and Eq. (23) can also be implemented in the DSP tocalculate the used thermal capacity dynamically. The relay shall trip ifthe used thermal capacity is greater than 1. Setting θ=1 in Eq. (17) andsolving for trip time gives: $\begin{matrix}{{t_{trip} = {T_{h}\ln\frac{M^{2} - \theta_{o}}{M^{2} - 1}}},} & 24 \\{{T_{h} = {{t_{trip} \div \ln}\frac{M^{2} - \theta_{o}}{M^{2} - 1}}},} & 25\end{matrix}$

Eq. (24) can be used to calculate trip time for different M when T_(h)is know while Eq. (25) can be used to calculate T_(h) when t_(trip) isknown. In the next section, Eq. (25) is used to derive the thermal timeconstants for the IEC standard trip classes. Note that the derivationfrom Eq. (1) to Eq. (18) is similar to the derivation in [6]. Furtherderivation is then different from [6] whose algorithm updates trip timeinstead of thermal capacity. Both standard and custom trip curves can bederived from Eq. (25). The derivations are demonstrated herein.

The derivations are first demonstrated on IEC standard trip classes. Thethermal time constants T_(h) and T_(c) in Eq. (22) and Eq. (23) usuallycan be obtained from motor manufacturer. In the exemplary case,discussed herein, however, IEC standard 60947-4-2 is used to derive thethermal time constants of IEC standard trip classes. Details of thederivation follow. For demonstration, the derivation is only performedon Class 10A, 10, 20, and 30. The derivation for other standard tripclasses can be obtained similarly. TABLE 2 Requirements of IEC 60947-4-2on performance of non- compensated overload relays Class 10A Class 10Class 20 Class 30 A (1.0 × I_(pickup)) t_(trip) > 2 h t_(trip) > 2 ht_(trip) > 2 h t_(trip) > 2 h B (1.2 × I_(pickup)) t_(trip) < 2 ht_(trip) < 2 h t_(trip) < 2 h t_(trip) < 2 h C (1.5 × I_(pickup))t_(trip) < 2 min t_(trip) < 4 min t_(trip) < 8 min t_(trip) < 12 min 7.2× I_(pickup) 2 < t_(trip) ≦10** 4 < t_(trip) ≦ 10** 6 < t_(trip) ≦ 20**9 < t_(trip) <= 30** 8 × I_(pickup) 1.6 ≦ t_(trip) ≦ 8.1* 3 ≦ t_(trip) ≦8.1* 5 ≦ t_(trip) ≦ 16.2* 7 ≦ t_(trip) ≦ 24.3* 7 × I_(pickup) 2 ≦t_(trip) ≦ 10.6* 4 ≦ t_(trip) ≦ 10.6* 6 ≦ t_(trip) ≦ 21.2* 9 ≦ t_(trip)≦ 31.7* 6 × I_(pickup) 3 ≦ t_(trip) ≦ 14.4* 6 ≦ t_(trip) ≦ 14.4* 9 ≦t_(trip) ≦ 28.8* 13 ≦ t_(trip) ≦ 43.2* 5 × I_(pickup) 4 ≦ t_(trip) ≦20.7* 8 ≦ t_(trip) ≦ 20.7* 12 ≦ t_(trip) ≦ 41.5* 19 ≦ t_(trip) ≦ 62.2* 4× I_(pickup) 6 ≦ t_(trip) ≦ 32.4* 13 ≦ t_(trip) ≦ 32.4* 19 ≦ t_(trip) ≦64.8* 29 ≦ t_(trip) ≦ 97.2* 3 × I_(pickup) 12 ≦ t_(trip) ≦ 57.6* 23 ≦t_(trip) ≦ 57.6* 35 ≦ t_(trip) ≦ 115.2* 52 ≦ t_(trip) ≦ 172.8* 2 ×I_(pickup) 26 ≦ t_(trip) ≦ 129.6* 52 ≦ t_(trip) ≦ 129.6* 78 ≦ t_(trip) ≦259.2* 112 ≦ t_(trip) ≦ 388.8*Note:1. All t_(trip) is counted from cold state at 40° C. except for B (1.2 ×I_(pickup)) the t_(trip) is counted from 2 h at 1.0 × I_(FLA) or fromrelay terminals reach thermal equilibrium at 1.0 × I_(FLA) whichever isless, and for C (1.5 × I_(pickup)) the t_(trip) is counted from thermalequilibrium at 1.0 × I_(FLA).2. All t_(trip) are in seconds except those stated otherwise.3. For M >= 2, only one set of tests need to be done. Preferably at 7.2× I_(pickup), i.e., the requirements denoted by ** are preferred tests.Multiple or single set of requirements denoted by * can be used asalternatives.

The above table summarizes the requirements of IEC on non-compensatedoverload relays. T_(h) can be calculated based on the requirement in theabove table using Eq. (25). For example, 7.2×I_(pickup) and Class 10A,2<t_(trip)<10, the middle value of $t_{trip} = {\frac{2 + 10}{2} = 6}$

sec, substituting t_(trip)=6 sec., M=7.2, and θ₀=0 into Eq. (25) givesthe thermal time constant T_(h)=308 sec. Because IEC prefers tests doneat 7.2×I_(pickup) and tests done at other M×I_(pickup) are justalternatives, the table below only shows the T_(h) calculated from7.2×I_(pickup). TABLE 3 Thermal time constants T_(h) calculated for M =7.2 and θ_(o) = 0 Class Class Class Class 10A 10 20 30 T_(h) 308 359 6671001 (sec)

The following demonstrates that the thermal algorithm may meet therequirements of IEC standards. The trip time for different M (note againthat $\left. {M = \frac{I}{I_{pickup}}} \right)$

can then be calculated using Eq. (24). Table 4, below, shows thecalculated t_(trip) for different M based on the T_(h) values in theabove Table 3. TABLE 4 Trip time t_(trip) calculated using the T_(h)values in Table 3 Class 10A Class 10 Class 20 Class 30 M = 8 4.85 5.6610.51 15.77 M = 7.2 6.00 7.00 13.00 19.50 M = 7 6.35 7.41 13.76 20.64 M= 6 8.68 10.12 18.80 28.20 M = 5 12.57 14.67 27.24 40.87 M = 4 19.8823.19 43.07 64.61 M = 3 36.28 42.33 78.61 117.91 M = 2 88.61 103.38192.00 288.00 M = 1.5 54.93* 64.08* 119.01* 178.51* M = 1.2 135.90*158.55* 294.45* 441.67* M = 1.05 731.72 853.67 1585.39 2378.08 M = 1 +1.0E−11 7588.41 8853.15 16441.56 24662.35Note:All t_(trip) is calculated from cold state at 40° C. except thatt_(trip) of M = 1.2* and M = 1.5* is calculated from thermal equilibriumat 1.0 × I_(FLA). SF is set to 1.15.

Observe that t_(trip) in above table 4 meets all the requirement shownin table 2. For demonstration, FIG. 8 shows the cold trip curves of IECclass 10A, 10, 20, and 30. Note that the hot trip curves and trip curvesof other IEC trip classes can be obtained similarly. The hot and coldtrip times for M=7.2 are listed below in Table 5. The hot trip time andcold trip time for M=7.2 can also be simulated using Eq. (22). TABLE 5Hot and cold trip time for M = 7.2 Class 10A Class 10 Class 20 Class 30Cold trip time (sec) 5.9994 6.9928 12.9922 19.4981 Hot trip time (sec)1.4738 1.7178 3.1916 4.7899

Simulation of IEC thermal memory test may also be accomplished. The newthermal memory test verification in [3] requires:

-   Applying a current equal to I_(pickup)±10% for two hours (equivalent    to the thermal equilibrium);-   interrupt the current for a duration of 2×T_(p)±10% (T_(p) is    defined in Table 2 of [3]);-   Apply a current equal to 7.2×I_(pickup);-   The relay shall trip within 50% of the cold trip time listed in    above Table 5.

FIGS. 3A and 3B show the simulation results for the IEC thermal memorytest verification. FIG. 3B shows an enlarged view around the point oft=7233 sec. Table 6 compares the IEC thermal memory test trip time tothe cold trip time. The IEC thermal memory test trip time is less than50% of the cold trip time. TABLE 6 Cold trip time and thermal memorytest trip time Class 10A Class 10 Class 20 Class 30 Cold trip time (sec)5.9994 6.9928 12.9922 19.4981 Thermal memory test 1.3 1.5 2.6 3.9 triptime (sec)

Examining the used thermal capacity for M<1, when motors are running atcurrent below pickup, the temperature rise at thermal equilibrium isproportional to the square of motor line current values. FIG. 4 showsthe used thermal capacity for five different M values.

For motors above 250 HP, manufacturers may provide thermal limit curvesthat specify the safe time for running overload and accelerationoverload. Curves 2 and 3 in FIG. 5 are the thermal limit curves. Onestandard trip class curve cannot provide required overload protectionfor both running overload and acceleration overload. For example, if astandard trip curve is selected based on acceleration safe time curve 3in FIG. 5, the trip time of the standard trip curve for running overload(the dashed line) is longer than the running safe time specified bymanufacturer (curve 2). Therefore a custom trip, for example, curve 1can be used, which means thermal time constant during acceleration isdifferent from that during running. Running thermal time constantT_(h-run) and acceleration thermal time constant T_(h-acc) can becalculated from the curves 2 and 3 using Eq. (25). Cooling thermal timeconstant T_(c)=3×T_(h-run)·M_(acc-run) is the multiple of pickup at thechanging point of thermal time constants.

FIGS. 6 and 7 demonstrate exemplary methods for applications using IECstandard or custom trip curves. In FIG. 6 a standard trip algorithm 600is provided. The algorithm 600 is initiated with the current at zero andthe thermal capacity at zero (block 602). The method determines if thecurrent is above zero (block 604). If the current is not above zero, themethod proceeds to initiation. If the current is above zero the motor isstarted, the thermal capacity is calculated using Eq. (22) and T_(h)values in table 3 (block 606). The method determines if the thermalcapacity is above one (block 608). If the thermal capacity is above one,the relay is tripped (block 610). If the thermal capacity remains belowor equal to one and the current is not equal to zero (block 612), themotor remains active and the thermal capacity is calculated using Eq.(22). If the relay is tripped or the motor is deactivated, the thermalcapacity is calculated using Eq. (23) and T_(c) equals 3×T_(h) (block614). Once the thermal capacity is back to zero (block 616), the methodis reinitiated (block 602).

In FIG. 7 a custom trip algorithm 700 is provided. The algorithm 700 isinitiated with the current at zero and the thermal capacity at zero(block 702). The method determines if the current is above zero (block704). If the current is not above zero, the method proceeds toinitiation. If the current is above zero, the motor is started and themethod determines if the status of the motor is active or in anacceleration process (block 706). If M_(n)≦M_(acc-run), the methoddetermines if the previous status of the motor was acceleration (block708). If the motor was accelerating, set status to run (block 710) andcalculate the thermal capacity using Eq. (22) with T_(h)=T_(h-run)(block 712). If the motor was not accelerating (block 708) and the motorwas run or active status (block 714), calculate the thermal capacityusing eq. 22 with T_(h)=T_(h-run) (block 712). If M_(n)>M_(acc-run), themethod determines if the previous status of the motor was run (block718). If the status was run, the method calculates the thermal capacityusing Eq. (22) with T_(h)=T_(h-acc) (block 716). If the status was notrun, the method sets the status to acceleration (block 720) andcalculates the thermal capacity using Eq. (22) with T_(h)=T_(h-acc)(block 716).

The method determines if the thermal capacity is above one (block 722).If the thermal capacity is above one, the relay is tripped (block 724).If the thermal capacity remains below or equal to one and the current isnot equal to zero (block 726), the motor remains active and the thermalcapacity is calculated as disclosed earlier in the algorithm 700. If therelay is tripped or the motor is deactivated, the thermal capacity iscalculated using Eq. (23) and T_(c) equals 3×T_(h-run) (block 728). Oncethe thermal capacity is back to zero (block 730), the method isreinitiated (block 702).

The systems and methods may be implemented using analog componentsand/or digital components. The systems and methods may be implementedwithin software that utilizes various components to implement theembodiments described herein. Aspects disclosed in the exemplaryembodiment may be utilized independently or in combination with otherexemplary embodiments. Moreover, it will be understood that theforegoing is only illustrative of the principles of the invention, andthat various modifications can be made by those skilled in the artwithout departing from the scope and spirit of the invention. Personsskilled in the art will appreciate that the present invention can bepracticed by other than the described embodiments, which are presentedfor purposes of illustration rather than of limitation, and the presentinvention is limited only by the claims that follow.

REFERENCES CITED

-   [1] IEEE Std C37.112-1996, IEEE Standard Inverse-Time Characteristic    Equations for Overcurrent Relays.-   [2] Benmouyal, G.; Meisinger, M.; Burnworth, J.; Elmore, W. A.;    Freirich, K.; Kotos, P. A.; Leblanc, P. R.; Lerley, P. J.;    McConnell, J. E.; Mizener, J.; Pinto de Sa, J.; Ramaswami, R.;    Sachdev, M. S.; Strang, W. M.; Waldron, J. E.; Watansiriroch, S.;    Zocholl, S. E.; “IEEE standard inverse-time characteristic equations    for overcurrent relays,” IEEE Transactions on Power Delivery, Volume    14, Issue 3, July 1999 Page(s): 868-872.-   [3] Amendment 2 to IEC 60947-4-2, Ed.2, Document No. 17B/1406/CC.-   [4] IEC 60947-4-2 Edition 2.1 2002-03, Low-Voltage Switchgear and    Controlgear—Part 4-2: Contactors and Motor-Starters—AC Semiconductor    Motor Controllers and Starters.-   [5] IEEE Std 620-1996, IEEE Guide for the Presentation of Thermal    Limit Curves for Squirrel Cage Induction Machines.-   [6] Abou-El-Ela, M. S.; Megahed, A. I.; Malik, O. P.; “Thermal model    based digital relaying algorithm for induction motor protection,”    Electrical and Computer Engineering, 1996. Canadian Conference on,    Volume 2, 26-29 May 1996 Page(s): 1016-1019 vol.2.-   [7] Samir F. Farag; T. Cronvich; “Motor Controller,” US patent    number US005206572A, Apr. 27, 1993.

1. An over-current relay protecting an induction motor comprising: amodule for determining the current drawn by the motor; a module fordetermining the state of the motor; a module for calculating a usedthermal capacity based on a first formula when the motor is in an activestate; a module for calculating the used thermal capacity based on asecond formula when the motor is in an inactive state; and a module fortripping the relay when the used thermal capacity attains a threshold.2. The relay of claim 1, wherein the first formula for used thermalcapacity is selected from the group consisting of$\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h}}} \right)}}$and  $\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}}}$3. The relay of claim 1, wherein the second formula for used thermalcapacity is selected from the group consisting of$\theta_{n} = {{\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{c}}} \right)\quad{and}\quad\theta_{n}} = {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{c}}}}}$4. The relay of claim 1, further comprising: a module for calculatingthe used thermal capacity based on a third formula when the motor is inan acceleration state.
 5. The relay of claim 4, wherein the thirdformula for used thermal capacity is selected from the group consistingof$\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h - {acc}}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h - {acc}}}} \right)}}$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {acc}}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {acc}}}}}}$6. The relay of claim 1, further comprising: a module for calculatingthe used thermal capacity based on a fourth formula when the motor is ina deceleration state.
 7. The relay of claim 6, wherein the fourthformula for used thermal capacity is selected from the group consistingof${\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h - {decel}}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h - {decel}}}} \right)}}}\quad$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {decel}}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {decel}}}}}}$8. The relay of claim 1, wherein the relay complies with InternationalElectrotechnical Commission standard 60947-4-2 and amendment 2 ofInternational Electrotechnical Commission standard 60947-4-2.
 9. Amethod for an over-current relay protecting an induction motorcomprising the following actions: determining the current drawn by themotor; determining the state of the motor; calculating a used thermalcapacity based on a first formula when the motor is in an active state;calculating the used thermal capacity based on a second formula when themotor is in an inactive state; and tripping the relay when the usedthermal capacity attains a threshold.
 10. The method of claim 9, whereinthe first formula for used thermal capacity is selected from the groupconsisting of$\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h}}} \right)}}$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}}}$11. The method of claim 9, wherein the second formula for used thermalcapacity is selected from the group consisting of$\theta_{n} = {{\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{c}}} \right)\quad{and}\quad\theta_{n}} = {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{c}}}}}$12. The method of claim 9, further comprising the following actions:calculating the used thermal capacity based on a third formula when themotor is in an acceleration state.
 13. The method of claim 12, whereinthe third formula for used thermal capacity is selected from the groupconsisting of${\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h - {acc}}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h - {acc}}}} \right)}}}\quad$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {acc}}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {acc}}}}}}$14. The method of claim 9, further comprising the following actions:calculating the used thermal capacity based on a fourth formula when themotor is in a deceleration state.
 15. The method of claim 9, wherein thefourth formula for used thermal capacity is selected from the groupconsisting of${\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h - {decel}}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h - {decel}}}} \right)}}}\quad$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{\frac{\Delta\quad t}{T_{h - {decel}}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h - {decel}}}}}}$16. The method of claim 14, wherein a standard IEC trip class curves isbased on induction motor thermal model IEC standard 60947-4-2 andamendment 2 of IEC standard 60947-4-2.
 17. An over-current relayprotecting an induction motor comprising: a module for determining thecurrent drawn by the motor; a module for determining the state of themotor; a module for calculating a used thermal capacity based on a firstformula when the motor is in an active state; a module for calculatingthe used thermal capacity based on a second formula when the motor is inan inactive state; and a module for tripping the relay when the usedthermal capacity attains a threshold.
 18. The relay of claim 17, whereinthe first formula for used thermal capacity is selected from the groupconsisting of${\theta_{n} = {{M_{n}^{2} \times \frac{\Delta\quad t}{T_{h}}} + {\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{h}}} \right)}}}\quad$and$\theta_{n} = {{M_{n}^{2}\left( {1 - {\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}} \right)} + {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{h}}}}}$19. The relay of claim 17, wherein the second formula for used thermalcapacity is selected from the group consisting of$\theta_{n} = {{\theta_{n - 1} \times \left( {1 - \frac{\Delta\quad t}{T_{c}}} \right)\quad{and}\quad\theta_{n}} = {\theta_{n - 1}{\mathbb{e}}^{- \frac{\Delta\quad t}{T_{c}}}}}$20. The system of claim 17, wherein the used thermal capacity tracks thethermal capacity throughout various motor states.